Linear Quadratic Stochastic Two-Person Nonzero-Sum Differential Games: Open-Loop and Closed-Loop Nash Equilibria

TL;DR

This paper analyzes linear quadratic stochastic two-person nonzero-sum differential games, characterizing Nash equilibria through Riccati equations and exploring the relationships between open-loop and closed-loop strategies.

Contribution

It provides a comprehensive characterization of open-loop and closed-loop Nash equilibria using Riccati equations and reveals their relationships in zero-sum cases.

Findings

Open-loop Nash equilibria characterized by forward-backward stochastic differential equations.

Closed-loop Nash equilibria characterized by coupled Riccati differential equations.

In zero-sum games, open-loop and closed-loop saddle points coincide through Riccati equations.

Abstract

In this paper, we consider a linear quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward-backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which is different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of open-loop saddle points coincides with that for the closed-loop saddle points, which leads to the conclusion that the closed-loop representation of open-loop saddle points is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear quadratic optimal control problem, the closed-loop representation of open-loop optimal controls coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.